IJPAM: Volume 56, No. 4 (2009)

$U$-CYCLES IN $N$-PERSON $TU$-GAMES WITH EQUAL-SIZED
OBJECTIONABLE FAMILIES OF COALITIONS

Juan C. Cesco$^1$, Ana L.Calí$^2$
$^1$Institute of Applied Mathematics of San Luis (IMASL)
CONICET - Universidad Nacional de San Luis
Av. Ejército de los Andes 950
San Luis, 5700, ARGENTINA
e-mail: jcesco@unsl.edu.ar
$^2$Department of Mathematics
Universidad Nacional de San Luis
Av. Ejército de los Andes 950
San Luis, 5700, ARGENTINA
e-mail: acali@unsl.edu.ar


Abstract.It has recently been proven that the non-existence of certain types of cycles of pre-imputation, fundamental cycles, is equivalent to the balancedness of a $TU$-game (see [3]). In some cases, the class of fundamental cycles can be narrowed and a characterization theorem may still be obtained. In this paper, we deal with $n$-person $TU$-games for which the only coalitions with non-zero value, aside from the grand coalition, are all coalitions of the same size $k\leq n,$ which also form a balanced family of coalitions. This class of games includes those studied in previous papers where the non-zero value coalitions are the family of coalitions with $n-1$ players. The main result obtained in this framework is that it is always possible to find a $U$-cycle, a certain type of fundamental cycle, provided the game under consideration is non-balanced and $n$ and $k$ are relatively prime. A computational procedure to get the cycle is provided as well. In many situations, these cycles turn out to be maximal $U$-cycles, an even more restricted class of fundamental cycles.

Received: August 25, 2006

AMS Subject Classification: 91A12

Key Words and Phrases: non-balanced games, cycles, transfer scheme

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 4